Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings
Computational Methods and Function Theory
We consider the convolution of half-plane harmonic mappings with respective dilatations (z+ a) / (1 + az) and eiθzn, where - 1 < a< 1 and θ∈ R, n∈ N. We prove that such convolutions are locally univalent for n= 1 , which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for n≥ 2.
Liu, Zhi Hong and Ponnusamy, Saminathan, "Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings" (2017). Journal Articles. 2544.